Immigration: Which consequences can arise for a company sponsoring an employee for a GreenCard in case of its further dissoluti?

It appears the website has classified your question to be one of employment law, but it would be better posed to immigration law attorneys. I wish...

How can I respond appropriate letter advocating for myself at work against some awful accusations from my supervisor? ?

Do you have a union? If so, consider grieving the PIP and working with your union representation. Also consider drafting a rebuttal for your file.

Sounds like you are doing nothing wrong. Be proactive and get out ahead of the problem though.

No, you cannot keep company property. Don't worry, if the employer refuses to turn over your paycheck, the NYS DOL wage enforcement division should...

Does an independent contractor, who works 45 hrs per week fixing truck, require the employer to pay over time?

That depends on if you are correctly classified as an independent contractor.

Is it legal for an employer to require you to cut your has as a male when the female employees have long hair flying all over?

If you contact the State, they may come in and require everyone to wear hair nets. I don't believe anyone is allowed to have long hair flow freely...

I was at work and a new employee was video snap chatting to a man in the woman's bathroom capturing me on video, is this illega?

In most States you have the right to be free of video recording in the restroom. Alert your employer right away.

It may come down to why they did this to you. If you feel there was some form of discrimination, I would speak with a local employment attorney...

"Can I"?

You may be out of luck, but I'd say it's worth scheduling a free consultation with a local attorney.

You may qualify for unemployment benefits if you resign.